Global existence for a fractionally damped nonlinear Jordan--Moore--Gibson--Thompson equation

TL;DR

This paper proves the global existence of solutions for a fractional damping nonlinear Jordan--Moore--Gibson--Thompson equation in nonlinear acoustics, addressing an open problem with minimal assumptions on the damping kernel.

Contribution

It establishes the global existence of solutions for the fractional damped JMGT model with quadratic gradient nonlinearity under minimal assumptions, including non-integrable kernels.

Findings

Global solutions exist under minimal damping assumptions

Addresses open problem for fractional damping JMGT models

Uses tailored analysis for non-integrable kernels

Abstract

In nonlinear acoustics, higher-order-in-time equations arise when taking into account a class of thermal relaxation laws in the modeling of sound wave propagation. In the literature, these families of equations came to be known as Jordan--Moore--Gibson--Thompson (JMGT) models. In this work, we show the global existence of solutions relying only on minimal assumptions on the nonlocal damping kernel. In particular, our result covers the until-now open question of global existence of solutions for the fractionally damped JMGT model with quadratic gradient nonlinearity. The factional damping setting forces us to work with non-integrable kernels, which require a tailored approach in the analysis to control. This approach relies on exploiting the specific nonlinearity structure combined with a weak damping provided by the nonlocality kernel.